Theorem isusp 22867

Description: The predicate 𝑊 is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)

Hypotheses

Ref	Expression
isusp.1	⊢ 𝐵 = (Base‘𝑊)
isusp.2	⊢ 𝑈 = (UnifSt‘𝑊)
isusp.3	⊢ 𝐽 = (TopOpen‘𝑊)

Assertion

Ref	Expression
isusp	⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))

Proof of Theorem isusp

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3459	. 2 ⊢ (𝑊 ∈ UnifSp → 𝑊 ∈ V)
2		0nep0 5223	. . . . 5 ⊢ ∅ ≠ {∅}
3		isusp.1	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑊)
4		fvprc 6638	. . . . . . . . . . . 12 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
5	3, 4	syl5eq 2845	. . . . . . . . . . 11 ⊢ (¬ 𝑊 ∈ V → 𝐵 = ∅)
6	5	fveq2d 6649	. . . . . . . . . 10 ⊢ (¬ 𝑊 ∈ V → (UnifOn‘𝐵) = (UnifOn‘∅))
7		ust0 22825	. . . . . . . . . 10 ⊢ (UnifOn‘∅) = {{∅}}
8	6, 7	eqtrdi 2849	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (UnifOn‘𝐵) = {{∅}})
9	8	eleq2d 2875	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 ∈ {{∅}}))
10		isusp.2	. . . . . . . . . 10 ⊢ 𝑈 = (UnifSt‘𝑊)
11	10	fvexi 6659	. . . . . . . . 9 ⊢ 𝑈 ∈ V
12	11	elsn 4540	. . . . . . . 8 ⊢ (𝑈 ∈ {{∅}} ↔ 𝑈 = {∅})
13	9, 12	syl6bb 290	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 = {∅}))
14		fvprc 6638	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
15	10, 14	syl5eq 2845	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → 𝑈 = ∅)
16	15	eqeq1d 2800	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (𝑈 = {∅} ↔ ∅ = {∅}))
17	13, 16	bitrd 282	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ = {∅}))
18	17	necon3bbid 3024	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (¬ 𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ ≠ {∅}))
19	2, 18	mpbiri 261	. . . 4 ⊢ (¬ 𝑊 ∈ V → ¬ 𝑈 ∈ (UnifOn‘𝐵))
20	19	con4i 114	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝐵) → 𝑊 ∈ V)
21	20	adantr 484	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)) → 𝑊 ∈ V)
22		fveq2 6645	. . . . . 6 ⊢ (𝑤 = 𝑊 → (UnifSt‘𝑤) = (UnifSt‘𝑊))
23	22, 10	eqtr4di 2851	. . . . 5 ⊢ (𝑤 = 𝑊 → (UnifSt‘𝑤) = 𝑈)
24		fveq2 6645	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
25	24, 3	eqtr4di 2851	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
26	25	fveq2d 6649	. . . . 5 ⊢ (𝑤 = 𝑊 → (UnifOn‘(Base‘𝑤)) = (UnifOn‘𝐵))
27	23, 26	eleq12d 2884	. . . 4 ⊢ (𝑤 = 𝑊 → ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ↔ 𝑈 ∈ (UnifOn‘𝐵)))
28		fveq2 6645	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
29		isusp.3	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑊)
30	28, 29	eqtr4di 2851	. . . . 5 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
31	23	fveq2d 6649	. . . . 5 ⊢ (𝑤 = 𝑊 → (unifTop‘(UnifSt‘𝑤)) = (unifTop‘𝑈))
32	30, 31	eqeq12d 2814	. . . 4 ⊢ (𝑤 = 𝑊 → ((TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)) ↔ 𝐽 = (unifTop‘𝑈)))
33	27, 32	anbi12d 633	. . 3 ⊢ (𝑤 = 𝑊 → (((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤))) ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
34		df-usp 22863	. . 3 ⊢ UnifSp = {𝑤 ∣ ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)))}
35	33, 34	elab2g 3616	. 2 ⊢ (𝑊 ∈ V → (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
36	1, 21, 35	pm5.21nii 383	1 ⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3441  ∅c0 4243  {csn 4525  ‘cfv 6324  Basecbs 16475  TopOpenctopn 16687  UnifOncust 22805  unifTopcutop 22836  UnifStcuss 22859  UnifSpcusp 22860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-res 5531  df-iota 6283  df-fun 6326  df-fv 6332  df-ust 22806  df-usp 22863

This theorem is referenced by:  ressust  22870  ressusp  22871  tususp  22878  uspreg  22880  ucncn  22891  neipcfilu  22902  ucnextcn  22910  xmsusp  23176